L(s) = 1 | + (−0.951 + 0.309i)3-s + 1.50i·7-s + (0.809 − 0.587i)9-s + (4.99 + 3.62i)11-s + (−2.09 − 2.87i)13-s + (0.471 + 0.153i)17-s + (−0.0963 + 0.296i)19-s + (−0.464 − 1.43i)21-s + (1.79 − 2.47i)23-s + (−0.587 + 0.809i)27-s + (0.0378 + 0.116i)29-s + (−0.909 + 2.79i)31-s + (−5.87 − 1.90i)33-s + (2.56 + 3.53i)37-s + (2.87 + 2.09i)39-s + ⋯ |
L(s) = 1 | + (−0.549 + 0.178i)3-s + 0.568i·7-s + (0.269 − 0.195i)9-s + (1.50 + 1.09i)11-s + (−0.579 − 0.797i)13-s + (0.114 + 0.0371i)17-s + (−0.0220 + 0.0680i)19-s + (−0.101 − 0.312i)21-s + (0.375 − 0.516i)23-s + (−0.113 + 0.155i)27-s + (0.00701 + 0.0216i)29-s + (−0.163 + 0.502i)31-s + (−1.02 − 0.332i)33-s + (0.421 + 0.580i)37-s + (0.460 + 0.334i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.362350634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362350634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.50iT - 7T^{2} \) |
| 11 | \( 1 + (-4.99 - 3.62i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.09 + 2.87i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.471 - 0.153i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.0963 - 0.296i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.79 + 2.47i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0378 - 0.116i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.909 - 2.79i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.56 - 3.53i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.44 - 2.50i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.62iT - 43T^{2} \) |
| 47 | \( 1 + (5.02 - 1.63i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (8.17 - 2.65i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.4 + 7.57i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.15 - 6.65i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-12.5 - 4.09i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.00 - 3.10i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.38 - 12.9i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.63 - 8.11i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (10.7 + 3.50i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.4 - 8.30i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 3.54i)T + (78.4 - 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.763985039704070740259555910464, −8.958519563059543278827796288235, −8.074823256127552587143323890102, −6.99781177714905507309592560986, −6.48493967268864172877580915487, −5.43981839902351401799268895309, −4.71489985670199054181835538022, −3.78193736930425394556797754597, −2.53081633459805012634376074482, −1.22879827767855654385783666959,
0.67090103300954349375175264491, 1.87287024447471654295577369256, 3.45505639259026150890515087717, 4.18510482213983580967170226571, 5.22306593968669199813705658688, 6.18585229934598258744933332108, 6.81855179559749303208930051973, 7.53767379403188275939726334037, 8.634734917379190452123726116108, 9.323962117093687014518094628014